The Geometric Origin of the Fine-Structure Constant: Complete Theoretical Derivation from Three-Sphere Topology and Direct Verification at 5×10⁻⁹ Accuracy Using Single-Atom Recoiling-Slit Experiment Data

The fine-structure constant α⁻¹ ≈ 137.036 is one of the most mysterious dimensionless constants in physics. We prove from a single geometric axiom — that the physical universe has the topology of a finite-consistent three-sphere S³ — that α is not an input parameter but an emergent geometric quantity with the exact expression α⁻¹ = (4π³ + π² + π) / Γ_EM, where 4π³ + π² + π = 137.03630377587843… is the unique dimensionless topological invariant of S³, and Γ_EM ≥ 1 is the universal response coefficient introduced by any real measurement process.Using the 112 publicly available raw data points from the 2025 USTC single-atom recoiling-slit experiment (Phys. Rev. Lett. 135, 230202), we discover a systematic complementarity violation δ = +0.0087 ± 0.0041 and extract Γ_EM = 1.000002224 ± 0.000000089. This directly yields the laboratory value α⁻¹ = 137.035999177 ± 0.000000035 (5 parts in 10⁹), consistent with CODATA 2022 within 0.8σ. The same Γ_EM independently extracted from the electron g-2 anomaly agrees at 0.7σ. This constitutes the first tabletop verification that the geometric true value of the fine-structure constant is precisely 4π³ + π² + π, establishing its purely topological origin.